Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C32×#41 ⋊ C2 |
| Sylow | C32 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#638 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C22×C3⋊D3 |
| Fitting | C32×#41 | C2 |
| socle | C2×C32 | C2×D4 |
| hypercenter | #41 | C3⋊D3 |
| hypocenter | C32 | AGL(1,ℤ/8) |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #41 | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,8)(4,6)(5,7)(12,13,14);(2,6)(5,8)(9,10,11);(2,5)(3,7)(6,8)(10,11)(13,14)Polycyclic presentation:
2^5,3^2|d^2=e,c^a=cd,d^a=de,f^a=f^2,g^a=g^2,c^b=ce,d^c=de51 conjugacy classes — view the full character table.
| maximal subgroup of | #10591666, #10591677, #10591680, … |
| maximal normal subgroup of | #10591666, #10591677, #10591680, … |
| maximal quotient of | #10592694, #10592700, #10592702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 806);